We note that in the proof, the constant \(K\)  is given in terms of the first eigenvalue of \(-A\) . The estimate (REF ) is commonly referred to as the logarithmic convexity of the solution to (). The logarithmic convexity method has been extensively investigated for backward parabolic problems, see for instance
[1]}, [2]}, [3]}, [4]}, [5]}, [6]} and the references therein. The method in [4]} relies on the complex variables to prove (REF ). Owing to the rapidly increasing references, we do not create a list of works and we refer to [8]} and [9]} for example.
However, to the best of the authors' knowledge, Theorem REF  is the first logarithmic convexity result for time fractional evolution equations.
